Let $g\in C([0,1]\times\mathbb{R}^n,\mathbb{R}^n)$ be locally Lipschitz in the second argument and bounded, and consider the IVP $$\dot{x}(t) = g(t,x(t)),\, t\in [0,1],\quad x(0) = x_0 \tag*{$(\star)_{x_0}$}.$$ This IVP has a unique solution $u_{x_0}$ for all $x_0\in\mathbb{R}^n$.
Now, consider some set $X$. What can we say about the set of final states of the solutions to $(\star)_{x_0}$ for $x_0\in X$, i.e. the set $$Y = \{u_{x_0}(1)\colon x_0\in X\}?$$ What properties of $X$ are invariant under $\left(X\to Y, x\mapsto u_{x}(1)\right)$?
E.g., if $X$ is bounded, one can show $Y$ is bounded, too. Or, what about $Y$ if $X$ is a relatively compact $k$-dimensional submanifold? Or...?
Thanks in advance.